Law of identity

In logic, the law of identity states that each thing is identical with itself. It is the first of the historical three laws of thought, along with the law of noncontradiction, and the law of excluded middle. However, few systems of logic are built on just these laws.

Ancient philosophy
The earliest recorded use of the law appears in Plato's dialogue Theaetetus (185a), wherein Socrates attempts to establish that what we call "sounds" and "colours" are two different classes of thing:

"Socrates: With regard to sound and colour, in the first place, do you think this about both: that they both are? Theaetetus: Yes. Socrates: Then do you think that each differs to the other, and the same as itself? Theaetetus: Certainly. Socrates: And that both are two and each of them one? Theaetetus: Yes, that too."

It is used explicitly only once in Aristotle, in a proof in the Prior Analytics:

"When A belongs to the whole of B and to C and is affirmed of nothing else, and B also belongs to all C, it is necessary that A and B should be convertible: for since A is said of B and C only, and B is affirmed both of itself and of C, it is clear that B will be said of everything of which A is said, except A itself."

- Aristotle

Medieval philosophy
Aristotle believed the law of non-contradiction to be the most fundamental law. Both Thomas Aquinas (Met. IV, lect. 6) and Duns Scotus (Quaest. sup. Met. IV, Q. 3) follow Aristotle in this respect. Antonius Andreas, the Spanish disciple of Scotus (d. 1320), argues that the first place should belong to the law "Every Being is a Being" (Omne Ens est Ens, Qq. in Met. IV, Q. 4), but the late scholastic writer Francisco Suárez (Disp. Met. III, § 3) disagreed, also preferring to follow Aristotle.

Another possible allusion to the same principle may be found in the writings of Nicholas of Cusa (1431–1464) where he says: "... there cannot be several things exactly the same, for in that case there would not be several things, but the same thing itself. Therefore all things both agree with and differ from one another."

Modern philosophy
Gottfried Wilhelm Leibniz claimed that the law of identity, which he expresses as "Everything is what it is", is the first primitive truth of reason which is affirmative, and the law of noncontradiction is the first negative truth (Nouv. Ess. IV, 2, § i), arguing that "the statement that a thing is what it is, is prior to the statement that it is not another thing" (Nouv. Ess. IV, 7, § 9). Wilhelm Wundt credits Gottfried Leibniz with the symbolic formulation, "A is A". Leibniz's Law is a similar principle, that if two objects have all the same properties, they are in fact one and the same: Fx and Fy iff x = y.

John Locke (Essay Concerning Human Understanding IV. vii. iv. ("Of Maxims") says:

"[...] whenever the mind with attention considers any proposition, so as to perceive the two ideas signified by the terms, and affirmed or denied one of the other to be the same or different; it is presently and infallibly certain of the truth of such a proposition; and this equally whether these propositions be in terms standing for more general ideas, or such as are less so: e.g., whether the general idea of Being be affirmed of itself, as in this proposition, "whatsoever is, is"; or a more particular idea be affirmed of itself, as "a man is a man"; or, "whatsoever is white is white" [...]"

Afrikan Spir proclaims the law of identity as the fundamental law of knowledge, which is opposed to the changing appearance of the empirical reality.

George Boole, in the introduction to his treatise The Laws of Thought made the following observation with respect to the nature of language and those principles that must inhere naturally within them, if they are to be intelligible:

"There exist, indeed, certain general principles founded in the very nature of language, by which the use of symbols, which are but the elements of scientific language, is determined. To a certain extent these elements are arbitrary. Their interpretation is purely conventional: we are permitted to employ them in whatever sense we please. But this permission is limited by two indispensable conditions, first, that from the sense once conventionally established we never, in the same process of reasoning, depart; secondly, that the laws by which the process is conducted be founded exclusively upon the above fixed sense or meaning of the symbols employed."

Objectivism, the philosophy founded by novelist Ayn Rand, is grounded in three axioms, one of which is the law of identity, "A is A". In the Objectivism of Ayn Rand, the law of identity is used with the concept existence to deduce that that which exists is something. Logic in Objectivist epistemology is based on the three laws of logic.

Analytic
In the Foundations of Arithmetic, Gottlob Frege associated the number one with the property of being self identical. Frege's paper "On Sense and Reference" begins with a discussion on equality and meaning. Frege wondered how a true statement of the form "a = a", a trivial instance of the law of identity, could be different from a true statement of the form "a = b", a genuine extension of knowledge, if the meaning of a term was its referent.

Bertrand Russell in "On Denoting" has this similar puzzle: "If a is identical with b, whatever is true of the one is true of the other, and either may be substituted for the other without altering the truth or falsehood of that proposition. Now George IV wished to know whether Scott was the author of Waverley; and in fact Scott was the author of Waverley. Hence we may substitute “Scott” for “the author of Waverley” and thereby prove that George IV wished to know whether Scott was Scott. Yet an interest in the law of identity can hardly be attributed to the first gentleman of Europe.”

In his "Tractatus Logico-Philosophicus", Ludwig Wittgenstein writes that "roughly speaking: to say of two things that they are identical is nonsense, and to say of one thing that it is identical with itself is to say nothing".

In the formal logic of analytical philosophy, the law of identity is written "a = a" or "For all x: x = x", where a or x refer to a term rather than a proposition, and thus the law of identity is not used in propositional logic. It is that which is expressed by the equals sign "=", the notion of identity or equality.

Continental
Martin Heidegger gave a talk in 1957 entitled "Der Satz der Identität" (The Statement of Identity), where he linked the law of identity "A=A" to the Parmenides' fragment "to gar auto estin noien te kai einai" (....for the same thing can be thought and can exist). Heidegger thus understands identity starting from the relationship of Thinking and Being, and from the belonging-together of Thinking and Being. Gilles Deleuze wrote that "Difference and Repetition" is prior to any concept of identity.

Modern logic
In first-order logic, identity (or equality) is represented as a two-place predicate, or relation, =. Identity is a relation on individuals. It is not a relation between propositions, and is not concerned with the meaning of propositions, nor with equivocation. The law of identity can be expressed as $$\forall x (x = x)$$, where x is a variable ranging over the domain of all individuals. In logic, there are various different ways identity can be handled. In first-order logic with identity, identity is treated as a logical constant and its axioms are part of the logic itself. Under this convention, the law of identity is a logical truth.

In first-order logic without identity, identity is treated as an interpretable predicate and its axioms are supplied by the theory. This allows a broader equivalence relation to be used that may allow a = b to be satisfied by distinct individuals a and b. Under this convention, a model is said to be normal when no distinct individuals a and b satisfy a = b.

One example of a logic that rejects or restricts the law of identity in this way is Schrödinger logic.